Limits

Here you will learn how to solve or evaluate limits at infinity with examples. Let’s begin – Limits at Infinity Algorithm to evaluate limits at infinity : 1). Write down the given expression in the form of a rational function i.e. $f(x)\over g(x)$, if it is not so. 2). If k is the highest power […]

Here you will learn what is the rationalisation method to solve or find limits with examples. Let’s begin – Rationalisation Method to Solve Limits This method is particularly used when either the numerator or denominator or both involve expression consisting of square roots and substituting the value of x the rational expression takes the form

Here you will learn what is the factorisation method to solve limits with examples. Let’s begin – Factorisation Method to Solve Limits Consider the following limit : $\displaystyle{\lim_{x \to a}}$ $f(x)\over g(x)$ If by substituting x = a, $f(x)\over g(x)$, reduces to the form $0\over 0$, then (x – a) is a factor of f(x)

Here you will learn direct substitution method to solve limits with examples. Let’s begin – Direct Substitution Method to Solve Limits Consider the following limits : (i)  $lim_{x \to a}$ f(x) (ii)  $lim_{x \to a} {\phi(x)\over \psi(x)}$ If f(a) and $\phi(a)\over \psi(a)$ exist and are fixed real numbers, then we say that $lim_{x \to a}$

Here you will learn some limits examples for better understanding of limit concepts. Example 1 : If $\displaystyle{\lim_{x \to \infty}}$(${x^3+1\over x^2+1}-(ax+b)$) = 2, then find the value of a and b. Solution : $\displaystyle{\lim_{x \to \infty}}$(${x^3+1\over x^2+1}-(ax+b)$) = 2 $\implies$ $\displaystyle{\lim_{x \to \infty}}$$x^3(1-a)-bx^2-ax+(1-b)\over x^2+1$ = 2 $\implies$ $\displaystyle{\lim_{x \to \infty}}$$x(1-a)-b-{a\over x}+{(1-b)\over x^2}\over 1+{1\over x^2}$ =